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Decision Analysis

Decision Analysis. Here we study the situation where the probability of each state of nature is known. Let’s reconsider the example from last section.

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Decision Analysis

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  1. Decision Analysis Here we study the situation where the probability of each state of nature is known.

  2. Let’s reconsider the example from last section. Say a firm recently purchased some land is considering putting condominiums on the land. The firm will put up either a 30, 60 or 90 unit condo complex. The outcome to the firm is influenced by the following states of nature: the demand for condos may be strong or the demand may be weak. Let’s put the information into a table, with the profit payoff values under the various scenarios – note we put the alternatives from which the firm will choose one in the rows and the states of nature in the columns. Say the probability of a strong demand for condos is .8, or P(s1)= 0.8 and thus P(s2) = 0.2.

  3. Expected Monetary Value (EMV) The EMV for each alternative is a weighted sum of possible payoffs for each alternative, where the weights are the probability of occurrence of each outcome. Digression: simple average The average of the numbers 1 and 2 is (1+2)/2 or .5(1+2) or .5(1) + .5(2) So, the simple average you know and love is really just a weighted sum where each term is weighted equally. What are the weights if you have 3 numbers and you want the average? So the EMV is a type of average, where the weights are the probability of occurrence of each outcome.

  4. Digress II Say I have a coin that comes up heads 60% of the time and comes up tails 40% of the time. A game: The game is you can bet on the flip of the coin. If the coin comes up heads you will be given $1, but if it comes up tails you will have to hand over a dollar. If you decide to pay the game once you will either win or lose $1. But if you continue to play over and over again the expected value is what you can expect to happen to you on average, per time played. Here the expected value = .6(1) + .4(-1) = .2. So, you would expected to win 20 cents on average.

  5. In the context of decision making under risk, the alternative to pick is the one with the highest EMV (unless you have something like costs and you would pick the one with the lowest value). In our example, d3 has the highest EMV. Now, in this example we do not know which state of nature will occur, just the probability of each state. What would we gain if we had perfect information? Maybe someone would like to sell us information that would help us know exactly the state of nature. Before we would buy the information (and we would have to pay, wouldn’t we?) we could see that if we had a strong demand we would pick d3 and if we had a weak demand we would pick d1. So, if we had perfect information (and in the context of making many decisions like this one) we could calculate the EVwPI.

  6. EVwPI, or the expected value with perfect information, is a weighted sum of the best option under each state. In our example we have .8(20) + .2(7) = 17.4 Now, if we take EVwPI minus EMV of the highest option (really called the expected value without perfect information, EVwoPI), we have the gain we would obtain if we had perfect information. We call it the EVPI, or the expected value of perfect information. In our example we have 17.4 – 14.2 = 3.2. So we would not pay more than 3.20 for the information because that is all it is worth to us.

  7. Sensitivity analysis In our example, d3 was the best response. In sensitivity analysis we ask the question, “how would the answer change if values in the problem were different?” Let’s look at this in the context of change the probabilities of each state. In our example, if the probabilities are handled generically as P and (1-P) (where P is the prob of a strong demand) then we have d1 P(8) + (1-P)(7) d2 P(14) + (1-P)(5) d3 P(20) + (1-P)(-9) On the next screen there is a graph to show some details.

  8. The horizontal axis here is the probability of a strong demand and the vertical is the expected value of each alt. Alternative d1 is the best if the probability of a strong demand is low, d2 is best if prob is medium and d3 is best if the probability of strong demand is high.

  9. You will note in the graph that the expected value lines for d1 and 2 are the highest at low levels of probability for a strong demand. To find the exact probability where they meet we just set the two equations together and solve for P. P(8) + (1-P)(7) = P(14) + (1-P)(5) Or P(8) + 7 – P(7) = P(14) + 5 – P(5) Or P + 7 = 9P +5 Or 8P = 2 Or P = ¼ So, d1 is best up to a P = ¼. Then d2 is best. Set d2 and d3 equal to find when d3 becomes best. P(14) + (1-P)(5) = P(20) + (1-P)(-9) Or 9P + 5 = 29P – 9 Or 20P = 14 Or P= 14/20 = .7

  10. So, the decision is sensitive to what is the probability of a strong demand for condos. The decision is also sensitive to the payoff values. In the current example d3 is the best all. If S is the generic payout to d3 when the demand is strong the expected value of d3 is .8S + .2(-9). The next best alternative was d2 with an expected value of 12.2. So as long as d3 has a value of S so that the expected value is better than or equal to 12.2 then d3 would still be the best. To find that S set the expected value of d3 equal to 12.2 and solve for S: .8S +.2(-9) = 12.2, Or S = (12.2 +1.8)/.8 = 17.5. Thus S can be as low as 17.5 and d3 would be the best option. When conducting sensitivity analysis, you only look at one variable changing at a time (like P, or S).

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